Dynamical Systems and Chaos — 620-341
2003
Assignment 2
• Due: Tuesday April 29 @ 5pm.
• You only need to attach a plagiarism coversheet if you didn’t submit the
first assignment.
• Put in the assignment box in the Maths department.
• Please be neat! Many people handed in appallingly messy work for assignment 1. — I will deduct up to 50% for messy work.
1. Using the java applets on the course homepage, calculate the locations of the first 6
superstable orbits of:
• The logistic map Fµ (x) = µx(1 − x), and
• The sine map Sµ (x) = µ sin(πx)
A point x0 is a superstable stable n-cycle if it has (F n )0 (x0 ) = 0. Since both these
maps have a unique maximum at x = 1/2, the superstable n-cycles occur at µ values
where the n-cycle contains the point x = 1/2. See next page for an example.
Remember — these maps undergo period doubling bifurcations, so the first 6 superstable orbits will have periods 1, 2, 22, . . . 25 . It is important that your estimates are
very accurate — as many decimal places as you can get! Take your time and be careful.
You will need to set “Discard first” and “Show next” to around 100,000 in order to get
good results.
Call these µ values for the logistic and sine maps l0 , l1 , . . . l5 and s0 , s1 , . . . s5 (respectively). To check you have the right idea — l0 = 2 and s0 = 1/2. These numbers are
expected to have the following behaviour:
ln ≈ l∞ − Aδln
sn ≈ s∞ − Bδsn
Tabulate your data and estimate δl and δs . What do your results suggest?
Hint: If ln ≈ l∞ − Aδln then
ln − ln−1 ≈ Aδ n−1 (δ − 1)
ln − ln−1
≈ δ
ln−1 − ln−2
Assignment 2
1
Due Tuesday April 29th @ 5pm
Dynamical Systems and Chaos — 620-341
Assignment 2
2003
2
Due Tuesday April 29th @ 5pm
Dynamical Systems and Chaos — 620-341
2003
2. Let ΣN denote the space of sequences whose entries are the integers 0, 1, . . . , N − 1.
(a) For all s, t ∈ ΣN define
d[s, t] =
∞
X
|sk − tk |
k=0
Nk
Prove that this function is a metric on ΣN , and find the maximum distance
between any two points in ΣN .
(b) Let σN be the shift map on ΣN (defined in the usual way). How many fixed points
k
does σN have? How many points are fixed by σN
?
Assignment 2
3
Due Tuesday April 29th @ 5pm